I remember a version from about the early 1960s in Co. Armagh, Northern Ireland, which was nearly identical to that mentioned by "Alison" in May 2006 above in Belfast:

Skinny Malink, Melodeon Legs, Big Banana Feet
Went to the pictures and couldn't get a seat
When he got a seat, he fell right through the seat
Skinny Malink, Melodeon Legs, Big Banana Feet

At that time I probably hadn't heard of the Musical instrument called a Melodeon, and when working on a building site in the summers of 1970 and 1971 was told that an adjustable spanner was called a Malojan (though I don't know if this was spelled this way). If you adjust the spanner, the two ends become closer or further apart, which to me seemed to resonate with the legs of a skinny lightweight person not being very robust, so that he or she might collapse under their weight. However it may well be that the musical instrument is the correct word. I also remember a play on TV set I think in Scotland (possibly Glasgow) many years ago, in which some children called a man "Mr. Skinny Malink.", which I think was not a compliment.